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Title

concord-- Concordance correlation coefficient and associated measures, tests, and graphs

Syntax

concordvaryvarx[if] [in] [weight] [,by(byvar)summarylevel(#)ccc[(norefccc_options)]loa[(norefreglineloa_options)]qnormd[(qnormd_options)]]

Description

concordcomputes Lin's (1989, 2000) concordance correlation coefficient for agreement on a continuous measure obtained by two persons or methods. (The measure was introduced earlier by Krippendorff (1970).) The concordance correlation coefficient combines measures of both precision and accuracy to determine how far the observed data deviate from the line of perfect concordance (i.e., the line at 45 degrees on a square scatter plot). Lin's coefficient increases in value as a function of the nearness of the data's reduced major axis to the line of perfect concordance (the accuracy of the data) and of the tightness of the data about its reduced major axis (the precision of the data). The Pearson correlation coefficient, r, the bias-correction factor, C_b, and the equation of the reduced major axis are reported to show these components. Note that the concordance correlation coefficient, rho_c, can be expressed as the product of r, the measure of precision, and C_b, the measure of accuracy.

concordalso provides results for Bland and Altman's limits-of-agreement, "loa", procedure (1986). The loa, a data-scale assessment of the degree of agreement, is a complementary approach to the relationship-scale approach of Lin.Finally, two other results are reported:

1. The correlation between difference and mean. In one interpretation this is a test statistic for a null hypothesis of equal variances given bivariate normality (Pitman 1939; also see Snedecor and Cochran 1989, 192-193). Alternatively, it is an exploratory diagnostic. A value near zero implies concordance.

2. An F test of equality of means and variances. Note that this too assumes bivariate normality (Bradley and Blackwood 1989). See also Hsu (1940) and Reynolds and Gregoire (1991). Nonsignificance implies concordance.

The user provides the pairs of measurements for a single property as observations in variables

varyandvarx. Frequency weights may be specified and used. Missing values (if any) are deleted in a casewise manner.Various associated graphs may be obtained through options. See below for explanations of the options

ccc,loa, andqnormd.

Options

by(byvar)produces separate results for groups of observations defined bybyvar.

summaryrequests summary statistics.

level(#)sets the confidence level % for the CI; default isc(level).

cccrequests a graphical display of the data and the reduced major axis of the data. The reduced major axis or SD line goes through the intersection of the means and has slope given by the sign of Pearson's r and the ratio of the standard deviations. The SD line serves as a summary of the center of the data.

ccc()suboptionnorefsuppresses the reference line of perfect concordance, y=x.

ccc()may also be specified with other options, which should be options ofscatter. For example, the scheme may be changed by a call such asccc(scheme(lean1).

loarequests a graphical display of the loa, the mean difference, and the data presented as paired differences plotted against pair-wise means.

loa()suboptionnorefsuppresses the reference line of perfect average agreement, y=0.

loa()suboptionreglineadds a regression line to the loa plot fitting the paired differences to the pair-wise means.

loa()may also be specified with other options, which should normally be options ofscatter. For example, the reference line of perfect average agreement was generated asloa(yline(0, lstyle(refline))yscale(range(0)) ylabel(0, add)).

qnormdrequests a normal plot of differences.

qnormd()may also be specified with options, which should be options ofscatter. For example,qnormd(title(Normal plot of differences))adds a title to the graph.

CommentsLin (2000) reported typographical errors in his original 1989 paper that affected calculation of the standard error of rho_c. These corrections were included in

concordin version 2.2.7 (January 2002) when the erratum was brought to the attention of the program authors by Dr. Benjamin Littenberg. We thank Dr. Littenberg.Kevan Polkinghorne pointed out a problem with

loa(). Mark Marshall pointed out a problem withby()under Stata 9.Dunn (2004) contains a bibliography on related work. Cox (2004) discusses other graphical approaches to this and related problems. Cox (2006) discusses concordance correlation and other numerical and graphical methods for assessing agreement with various scientific examples.

Saved resultsThe following items are returned in

r(), if theby()option was not used:

r(N)number of observations comparedr(rho_c)concordance correlation coefficient rho_cr(se_rho_c)standard error of rho_cr(asym_ll)lower CI limit (asymptotic)r(asym_ul)upper CI limit (asymptotic)r(z_tr_ll)lower CI limit (z-transform)r(z_tr_ul)upper CI limit (z-transform)r(C_b)bias-correction factor C_br(diff)mean differencer(sd_diff)standard deviation of mean differencer(LOA_ll)lower loa CI limitr(LOA_ul)upper loa CI limitr(rdm)correlation between difference and meanr(Fdm)F from Bradley-Blackwood test

Examples

. concord rater1 rater2

. concord rater1 rater2 [fw=freq]

. concord rater1 rater2, summary ccc

. concord rater1 rater2, summary ccc(noref)

. concord rater1 rater2, level(90) by(grp)

. concord rater1 rater2, loa(regline noref)

. concord rater1 rater2, qnormd(title(Normal plot))

AuthorsThomas J. Steichen, Winston-Salem, NC, USA, steichen@triad.rr.com

Nicholas J. Cox, Durham University, UK, n.j.cox@durham.ac.uk

ReferencesBland, J. M., and D. G. Altman. 1986. Statistical methods for assessing agreement between two methods of clinical measurement.

LancetI: 307-310.Bradley, E. L., and L. G. Blackwood. 1989. Comparing paired data: a simultaneous test for means and variances.

American Statistician43: 234-235.Cox, N. J. 2004. Graphing agreement and disagreement.

Stata Journal4: 329-349.------. 2006. Assessing agreement of measurements and predictions in geomorphology.

Geomorphology76: 332-346.Dunn, G. 2004.

Statistical Evaluation of Measurement Errors: Design andAnalysis of Reliability Studies.London: Arnold.Hsu, C. T. 1940. On samples from a normal bivariate population.

Annalsof Mathematical Statistics11: 410-426.Krippendorff, K. 1970. Bivariate agreement coefficients for reliability of data. In Borgatta, E.F. and G.W. Bohrnstedt (eds)

SociologicalMethodology. San Francisco: Jossey-Bass, 139-150. [a.k.a.Sociological Methodology2: 139-150]Lin, L. I-K. 1989. A concordance correlation coefficient to evaluate reproducibility.

Biometrics45: 255-268.------. 2000. A note on the concordance correlation coefficient.

Biometrics56: 324-325.Pitman, E. J. G. 1939. A note on normal correlation.

Biometrika31: 9-12.Reynolds, M., and T. G. Gregoire. 1991. Comment on Bradley and Blackwood.

American Statistician45: 163-164.Snedecor, G. W., and W. G. Cochran. 1989.

Statistical Methods.Ames, IA: Iowa State University Press.

Also seeSTB: STB-43 sg84; STB-45 sg84.1; STB-54 sg84.2; STB-58 sg84.3

SJ: SJ2-2 st0015; SJ4-4 st0015_1; SJ5-3: st0015_2; SJ6-2: st0015_3